# Invertibility via distance for noncentered random matrices with continuous distributions

@article{Tikhomirov2020InvertibilityVD,
title={Invertibility via distance for noncentered random matrices with continuous distributions},
author={Konstantin E. Tikhomirov},
journal={Random Struct. Algorithms},
year={2020},
volume={57},
pages={526-562}
}
Let $A$ be an $n\times n$ random matrix with independent rows $R_1(A),\dots,R_n(A)$, and assume that for any $i\leq n$ and any three-dimensional linear subspace $F\subset {\mathbb R}^n$ the orthogonal projection of $R_i(A)$ onto $F$ has distribution density $\rho(x):F\to{\mathbb R}_+$ satisfying $\rho(x)\leq C_1/\max(1,\|x\|_2^{2000})$ ($x\in F$) for some constant $C_1>0$. We show that for any fixed $n\times n$ real matrix $M$ we have $${\mathbb P}\{s_{\min}(A+M)\leq t n^{-1/2}\}\leq C'\, t… Expand #### Topics from this paper Overlaps, Eigenvalue Gaps, and Pseudospectrum under real Ginibre and Absolutely Continuous Perturbations • Mathematics, Computer Science • ArXiv • 2020 A key ingredient in the proof is new lower tail bounds on the small singular values of the complex shifts z-(A+\gamma G_n) which recover the tail behavior of thecomplex Ginibre ensemble when \Im z\neq 0 yields sharp control on the area of the pseudospectrum \Lambda_\epsilon(A+ \gammaG_n). 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